US 8204232 B2 Abstract Accelerated computation of combinations of group operations in a finite field is provided by arranging for at least one of the operands to have a relatively small bit length. In a elliptic curve group, verification that a value representative of a point R corresponds the sum of two other points uG and vG is obtained by deriving integers w,z of reduced bit length and so that v=w/z. The verification equality R=uG+vQ may then be computed as −zR+(uz mod n) G+wQ=O with z and w of reduced bit length. This is beneficial in digital signature verification where increased verification can be attained.
Claims(32) 1. In a cryptographic module having an arithmetic processing unit, a method of verifying a relationship between a sum of scalar multiples of a pair of points on an elliptic curve and a third point on said curve, the sum of scalar multiples of said pair of points including a first scalar multiplied by a first point of said pair of points added to a second scalar multiplied by a second point of said pair of points, the method comprising:
the cryptographic module obtaining a pair of integers, each of the pair of integers having a bit length less than the bit length of one of said first scalar and said second scalar and wherein the ratio of the pair of integers corresponds to said one of said first scalar and said second scalar,
the arithmetic processing unit performing computations from which said relationship can be verified, said computations involving said first point, said second point, said third point, and said pair of integers, wherein said integers are used in said computations instead of said one of said first scalar and said second scalar, thereby reducing the number of said computations,
subsequent to said computations, the cryptographic module determining whether the result of said computations indicates verification of said relationship, and
if the result of said computations indicates the verification of said relationship, producing an output indicative of the verification.
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16. A method of verifying a digital signature of a message, said digital signature having been generated from said message by performing cryptographic operations in a group having elements represented by bit strings, said signature comprising a first signature component and a second signature component, the first signature component having been derived from an ephemeral public key of a signer and the second signature component having been derived from said message, said first signature component, an ephemeral private key of said signer, and a long term private key of said signer, said method comprising:
obtaining said ephemeral public key from said first signature component;
computing at least one of a pair of scalars, a first of which is computed using said message and said second signature component, and a second of which is computed using said first signature component and said second signature component;
obtaining a pair of integers, each of the pair of integers having a bit length less than the bit length of one of said scalars and wherein the ratio of the pair of integers corresponds to said one of said scalars;
performing computations from which said digital signature can be verified, said computations involving said ephemeral public key, a long term public key of the signer, and a generator of said group, wherein said pair of integers is used in said computations instead of said one of said scalars, thereby reducing the number of said computations
determining whether the result of said computations indicates verification of said digital signature; and
accepting said digital signature only if the result of said computation indicates the verification of said digital signature.
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R is the ephemeral public key;
G is the generator of the group;
Q is the long term public key of the signer;
n is the order of the group;
z, w are said integers
u is said first of said scalars; and
o is the group identity.
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^{−1}(H(M)+dr)mod n where n is an order of a generator point, H(M) is a secure hash of said message, and d is the long term private key of said signer, and wherein said first of said scalars u corresponds to H(M)/s mod n.22. The method according to
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^{i}G is computed and values from said table used in said performing said computations.27. The method according to
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31. A cryptographic module having an arithmetic processing unit, the cryptographic module operable to perform operations for verifying a relationship between a sum of scalar multiples of a pair of points on an elliptic curve and a third point on said curve, the sum of scalar multiples of said pair of points including a first scalar multiplied by a first point of said pair of points added to a second scalar multiplied by a second point of said pair of points, said operations comprising:
obtaining a pair of integers, each of the pair of integers having a bit length less than the bit length of one of said first scalar and said second scalar and wherein the ratio of the pair of integers corresponds to said one of said first scalar and said second scalar,
performing computations from which said relationship can be verified, said computations involving said first point, said second point, said third point, and said pair of integers, wherein said integers are used in said computations instead of said one of said first scalar and said second scalar, thereby reducing the number of said computations,
subsequent to said computations, determining whether the result of said computations indicates verification of said relationship, and
if the result of said computations indicates the verification of said relationship, producing an output indicative of the verification.
32. A cryptographic module operable to perform operations for verifying a digital signature of a message, said digital signature having been generated from said message by performing cryptographic operations in a group having elements represented by bit strings, said signature comprising a first signature component and a second signature component, the first signature component having been derived from an ephemeral public key of a signer and the second signature component having been derived from said message, said first signature component, an ephemeral private key of said signer, and a long term private key of said signer, said operations comprising:
obtaining said ephemeral public key from said first signature component;
computing at least one of a pair of scalars, a first of which is computed using said message and said second signature component, and a second of which is computed using said first signature component and said second signature component;
obtaining a pair of integers, each of the pair of integers having a bit length less than the bit length of one of said scalars and wherein the ratio of the pair of integers corresponds to said one of said scalars;
performing computations from which said digital signature can be verified said computations involving said ephemeral public key, a long term public key of the signer, and a generator of said group, wherein said pair of integers is used in said computations instead of said one of said scalars, thereby reducing the number of said computations
determining whether the result of said computations indicates verification of said digital signature; and
accepting said digital signature only if the result of said computation indicates the verification of said digital signature.
Description This application claims priority from U.S. Provisional Application No. 60/644,034 filed on Jan. 18, 2005, the content of which is hereby incorporated by reference. The present invention relates to computational techniques used in cryptographic algorithms. The security and authenticity of information transferred through data communication systems is of paramount importance. Much of the information is of a sensitive nature and lack of proper control may result in economic and personal loss. Cryptographic systems have been developed to address such concerns. Public key cryptography permits the secure communication over a data communication system without the necessity to transfer identical keys to other parties in the information exchange through independent mechanisms, such as a courier or the like. Public key cryptography is based upon the generation of a key pair, one of which is private and the other public that are related by a one way mathematical function. The one way function is such that, in the underlying mathematical structure, the public key is readily computed from the private key but the private key cannot feasibly be ascertained from the public key. One of the more robust one way functions involves exponentiation in a finite field where an integer k is used as a private key and the generator of the field α is exponentiated to provide a public key K=α The security of such systems is dependent to a large part on the underlying mathematical structure. The most commonly used structure for implementing discrete logarithm systems is a cyclic subgroup of a multiplicative group of a finite field in which the group operation is multiplication or cyclic subgroups of elliptic curve groups in which the group operation is addition. An elliptic curve E is a set of points of the form (x, y) where x and y are in a field F, such as the integers modulo a prime p, commonly referred to as Fp, and x and y satisfy a non-singular cubic equation, which can take the form y Scalar multiplication can be defined from addition as follows. For any point P and any positive integer d, dP is defined as P+P+ . . . +P, where d occurrences of P occur. Thus 1P=P and 2P=P+P, and 3P=P+P+P, and so on. We also define 0P=O and (−d)P=d(−P). For simplicity, it is preferable to work with an elliptic curve that is cyclic (defined below) although in practice, sometimes a cyclic subgroup of the elliptic curve is used instead. Being cyclic means that there is a generator G, which is a point in the group such that every other point P in the group is a multiple of G, that is to say, P=dG, for some positive integer d. The smallest positive integer n such that nG=O is the order of G (and of the curve E, when E is cyclic). In cryptographic applications, the elliptic curves are chosen so that n is prime. In an elliptic curve cryptosystem, the analogue to exponentiation is point multiplication. Thus it is a private key is an integer k, the corresponding public key is the point kP, where P is a predefined point on the curve that is part of the system parameters. The seed point P will typically be the generator G. The key pair may be used with various cryptographic algorithms to establish common keys for encryption and to perform digital signatures. Such algorithms frequently require the verification of certain operations by comparing a pair of values as to confirm a defined relationship, referred to as the verification equality, between a set of values. One such algorithm is the Elliptic Curve Digital Signature Algorithm (ECDSA) used to generate digital signatures on messages exchanged between entities. Entities using ECDSA have two roles, that of a signer and that of a verifier. A signer selects a long term private key d, which is an integer d between 1 and n−1 inclusive. The integer d must be secret, so it is generally preferable to choose d at random. The signer computes Q=dG. The point Q is the long term public key of the signer, and is made available to the verifiers. Generally, the verifiers will have assurance generally by way of a certificate from a CA, that Q corresponds to the entity who is the signer. Finding the private key d from the public key Q is believed to an intractable problem for the choices of elliptic curves used today. For any message M, the signer can create a signature, which is a pair of integers (r, s) in the case ECDSA. Any verifier can take the message M, the public key Q, and the signature (r, s), and verify whether it was created by the corresponding signer. This is because creation of a valid signature (r, s) is believed to possible only by an entity who knows the private key d corresponding to the public key Q. The signing process is as follows. First, the signer chooses some integer k in the interval [1, n−1] that is to be used as a session, or ephemeral, private key. The value k must be secret, so generally it is preferable to choose k randomly. Then, the signer computes a point R=kG that has co-ordinates (x, y). Next, the signer converts x to an integer x′ and then computes r=x′ mod n, which is the first coordinate of the signature. The signer must also compute the integer e=h(M) mod n, where h is some hash function, generally one of the Secure Hash Algorithms (such as SHA-1 or SHA-256) defined in Federal Information Processing Standard (FIPS) 180-2. Finally, the second coordinate s is computed as s=(e+dr)/s mod n. The components (r,s) are used by the signer as the signature of the message, M, and sent with the message to the intended recipient. The verifying process is as follows. First the verifier computes an integer e=h(M) mod n from the received message. Then the verifier computes integers u and v such that u=e/s mod n and v=r/s mod n. Next, the verifier computes a value corresponding to the point R that is obtained by adding u G+v Q. This has co-ordinates (x, y). Finally the verifier converts the field element x to an integer x′ and checks that r=x′ mod n. If it does the signature is verified. From the above, the verification of an ECDSA signature appears to take twice as long as the creation of an ECDSA signature, because the verification process involves two scalar multiplications, namely uG and vQ, whereas signing involves only one scalar multiplication, namely kG. Elliptic curve scalar multiplications consume most of the time of these processes, so twice as many of them essentially doubles the computation time. Methods are known for computing uG+vQ that takes less time than computing uG and vG separately. Some of these methods are attributed to Shamir, some to Solinas, and some to various others. Generally, these methods mean that computing uG+vQ can take 1.5 times as long as computing kG. Another commonly used method to accelerate elliptic curve computations is pre-computing tables of multiples of G. Such pre-computed tables save time, because the point G is generally a fixed system parameter that is re-used repeatedly. The simplest pre-compute table consists of all multiples 2^j G for j from 0 to t, where t is the bit-length of n. With such a pre-computed table, computing an arbitrary multiple kG can be done with an average of t/2 point additions or less. Roughly, this a threefold improvement over the basic method of computing kG, which clearly demonstrates the benefit of pre-computation. Generally speaking, larger pre-computed tables yield better time improvements. The memory needed to store the pre-computed tables has a significant cost. Therefore, implementers must balance the benefit of faster operations with the extra cost of larger tables. The exact balance generally depends of the relative importance of speed versus memory usage, which can vary from one implementation to another. Pre-computation can also be applied to the public key Q. Generally, the public key Q tends to vary more often than G: as it is different for each correspondent, whereas G is always fixed for a given system. Therefore the cost of one-time pre-computation for Q is amortized over a smaller number of repeated run-time computations involving Q. Nevertheless, if Q is to be used more than once, some net savings on time will be achieved. Public keys that are heavily used include those of certification authorities (CA), especially root, trusted or anchor CA public keys (that are pre-installed into a system). Therefore, pre-computation may be worthwhile for CA elliptic curve public keys where, for example, the protocol requires verification of a CA's certificate. Another difference between pre-computations of Q versus G is the cost of storing or communicating the pre-computed tables. Each public key Q requires its own pre-computed table. In a system with many distinct public keys, these costs may accumulate to the point that any benefit of faster computation is offset by the need to store or communicate keys. The net benefit depends on the relative cost of time, memory and bandwidth, which can vary tremendously between implementations and systems. Again, in the case of CA public keys, especially root, trusted or anchor CA keys, these keys tend to be fewer in number than end-entity public keys, so that the cost of pre-computation will generally be less and amortized over more operations. Tables of multiples of points are not merely useful during pre-computation. In practice, such tables are commonly generated at run-time, during an initial phase of each computation. The savings provided by these tables is essentially that of avoiding certain repetitious operations that occur within a single computation. A single computation has less internal repetitions than two distinct computations have in common, so that saved repetition amount to less than pre-computation. Nevertheless, it has been found that with a judicious choice of table, the time need for a single computation can be reduced. The table takes time to compute, and computation of the table cannot be amortized over multiple computations, so is incurred for every computation. Experience has shown that particular tables decrease the amount of time needed because computing the table takes less time than the repetition operations that would have otherwise been needed. Usually, there is an optimum size and choice of table. Another cost of such tables is the memory needed to temporarily store the table. The cost of such memory may affect the optimal choice of table. Windowing methods are examples of such tables computed on the fly. Not withstanding all of the above known techniques for efficient implementation, further efficiency improvements are desirable. In particular, the efficiency of verifying of ECDSA signatures is particularly desirable. Extensive pre-computation allows ECDSA signatures to be generated very quickly. In fact, ECDSA signature generation is one of the fastest digital signature generation algorithms known. On the other hand, ECDSA signature verification is relatively slower, and there are other signature algorithms have similar verification times to ECDSA. Improvement of ECDSA verification time is therefore important, especially for environments where verification time is a bottleneck. In general, there is a need to enhance the efficiency of performing a computation to verify that a value corresponds to the sum of two of the values. It is therefore an object of the present invention to obviate or mitigate the above disadvantages. In general terms the present invention provides a method and apparatus for verifying the equality of a relationship between the sum of scalar multiples of a pair of points on an elliptic curve and a third point on said curve. The method comprises the steps of i) obtaining a pair of integers of bit length less than one of said scalars and whose ratio corresponds to said scalar; ii) substituting said integers for said scalars in said relationship to obtain an equivalent relationship in which at least one of said terms is a scalar multiple of one of said points with reduced bit length, and iii) computing said equivalent relationship to verify said equality. The method may be used for verifying that a value representative of a point R on an elliptic curve corresponds to the sum of two other points, uG and vQ. Integers w and z are determined such that the bit lengths of the bit strings representing w and z are each less than the bit length of the bit string of the integer v, and such that v=w/z mod n. With such w and z, the equation R=uG+vQ can be verified as −zR+(zu mod n)G+wQ=O. Preferably, the bit lengths of w and z are each about half the bit length of n, which means that both w and z are both no larger than about n The point −zR+(zu mod n)G+wQ can be computed efficiently because z and w are relatively small integers, and various of the methods for computing a sum faster than its parts can be used. The multiple (zu mod n) is full size, but within the context of an algorithm such as the ECDSA, the point G may be fixed or recurring. In this case the computation can be accelerated with the use of a stored table for G. Estimates of the times savings for this approach compared to conventional verification with tables for G are around 40%. The values w and z may be obtained by using a partial completed extended Euclidean algorithm computation. Alternatively, a continued fractions approach may be utilised to obtain w and z efficiently. In a further aspect of the invention there is provided a method of verifying a digital signature of a message performed by a cryptographic operation in a group of a finite field having elements represented by bit strings of defined maximum bit length. The signature comprises a pair of components, one of which is derived from an ephemeral public key of a signer and the other of which combines the message, the first component and the ephemeral public key and a long term public key of the signer. The method comprises the steps of recovering the ephemeral public key from the first component, establishing a verification equality as a combination of group operations on the ephemeral public key, the long term public key and a generator of the group with at least one of the group operations involving an operand represented by bit strings having a reduced bit length less than the defined maximum bit length, computing the combination and accepting the signature if said equality holds and rejecting the signature if said equality fails. Preferably, the group is a elliptic curve group. As a further aspect, a method of generating a signature of a message by a cryptographic operation in an elliptic curve group of finite field comprising the steps of generating a pair of signature components with one of said components derived from a point representing an ephemeral public key and including in said signature an indicator to identify one of a plurality of possible values of said public key that may be recovered from said one component. Embodiments of the invention will now be described by way of example only with reference to the accompanying drawings in which: The present invention is exemplified by reference to verification of digital signatures, in particular those signatures generated using ECDSA. It will be apparent however that the techniques described are applicable to other algorithms in which verification of a pair of values representative of points on an elliptic curve is required to groups other than elliptic curve groups. Therefore the accompanying description of preferred embodiments is exemplary and not exhaustive. Referring therefore to In the present example, the correspondent The steps taken to sign the message are shown in In addition to the components r and s, the signature includes information i to permit the co-ordinates representing the point R to be recovered from the component r. This information may be embedded in the message M, or forwarded as a separate component with r and s and will be used by the verifier to compute the value R If the elliptic curve is defined over a field F of prime order p, and the elliptic curve E is cyclic or prime order n, then i can generally be taken as y mod 2, i.e., a zero or one. The indication i is required during recovery R, where the verifier sets x=r. It is very likely that x=r because n and p are extremely close for typical implementations. Given x, there are exactly two values y such that (x, y) is on the curve, and these two values y and y′ have different values mod 2. Thus i is just a single bit whose value indicates which of the y's is to be used, and adds relatively little cost to the signature. Once the message is signed it is forwarded together with the components r,s, and i across the link The correspondent Thus, the arithmetic unit Let r Let r For i>1, determine ri, ti as follows: Use the division algorithm to write r_(i−1)=qi r_(i−2)+ri, which defines ri.
Stop as soon as ri<sqrt(n)=n^(½), or some other desired size. Set w=ri and z=ti. Note that ri=ti v mod n, so w=z v mod n, so v=w/z mod n, and both w and z have about half the bit length of n, as desired. The correspondent With the value of R recovered, the verification of the ECDSA, namely that R=uG+vQ, may proceed with a revised verification by confirming that the verification equality −zR+(zu mod n)G+wQ=O. The verification equality −zR+(zu mod n)G+wQ involves a combination of group operations on each of the ephemeral public key R, generator G and long term public key Q and can be computed efficiently because z and w are relatively small integers. As will be described below, various of the methods for computing a sum faster than its parts can be used. The multiple (zu mod n) is full size, but within the context of a system such as the ECDSA in which the points have varying longevity, the point G may be considered fixed or recurring. In this case the computation can be accelerated with a precomputed table for G, which may be stored in the memory A number of different known techniques may be utilised to compute the required relationship, each of which may be implemented using the arithmetic processor The revised verification instead uses a computation of a combination of the form aG+wQ+zR, where a is an integer of bit length t representative of the value zu mod n and w and z are integers of bit length about (t/2). Organising the verification computation in this way permits a number of efficient techniques to be used to reduce the number of point operations. An efficient way to compute this is to use a simultaneous doubling and add algorithm. For example, if the relationship 15G+20Q+13R is to be computed it can be done in stages as 2Q; G+2Q; G+2Q+R; 2G+4Q+2R; 3G+4Q+2R; 3G+5Q+2R; 3G+5Q+3R; 6G+10Q+6R; 7G+10Q+6R; 14G+20Q+12R; 15G+20Q+13R, for a total of 12 point additions, which is fewer than the method of generic scalar multiplication for each term separately. The main way that this method uses less operations is when it does simultaneous doubling, in steps as going from G+2Q+R to 2G+4Q+2R. In computing each term separately three operations would be used corresponding to this one operation. In fact, three simultaneous doubling were used, each saving two operations, so simultaneous doubling account precisely for all the savings. The number of doublings to compute the combination is governed by the length of the highest multiple, so it is t. The number of additions for a is (t/2), on average, and for Q and R it is (t/4) each on average. The total, on average, is t+(t/2)+(t/4)+(t/4)=2t. The algorithm is further exemplified as Algorithm 3.48 of the Guide to Elliptic Curve Cryptography detailed above. Although there does not appear to be any savings over the previous method, which also took 2t point operations, advantage can be taken of the fact that in practice, for ECDSA, the generator G is constant. This allows the point J=2^m G to be computed in advance, and stored in memory Accordingly, to verify the signature r,s, as shown schematically in With the conventional verification equation approach of computing uG+vQ, the multiple v will generally be full length t, so the pre-computed multiple J of G will not help reduce the number of simultaneous doublings. Therefore, by pre-computing and storing the single point J, verifying using the relationship −zR+(zu mod n)G+wQ=O allows an ECDSA signature to be verified in 25% less time. In other words, 33% more signatures can be verified in a given amount of time using the embodiment described above. Alternatively, many implementations have sufficient memory When signed binary expansions are used, then computing uG+vQ (without any pre-computation) requires about t doublings and (t/3) additions for each of G and Q, for a total of (10/6)t operations, on average. When signed binary expansions are used to find a′G+a″J+wQ+zR, about t/2 doublings are needed, and (t/6) additions for each of G, J, Q and R, for a total of (7/6)t operations, on average. The time to verify using the verification representation described above is 70% compared to without, or 30% less. This allows about 42% more signatures to verified in a given amount of time. The advantage of verifying using the revised verification representation is increased when combined with a more advanced technique of scalar multiplication referred to as signed binary expansions. This technique is very commonly used today in elliptic curve cryptography, so today's existing implementations stand to benefit from adoption of the verification representations. Accordingly, it will be seen that by reorganizing the verification equation so that signature variables have a reduced bit length, the speed of verification may be increased significantly. In the above embodiments, the recipient performs computations on the components r,s. To further accelerate signature verification as shown in Upon receipt, the verifier computes w and z. The verifier then determines c=c′+c″β and b=b′+b″β+b′″β^2. In addition, since G is a fixed parameter, the verifier has pre-computed multiples of G of the form βG and β^2G. If n is approximately 2^t, then the verifier needs just t/3 simultaneous doubles to compute aR+bG+cQ. The verification can proceed on the basis aR+(b′+b″β+b′″β^2)G+(c′+c″β)Q=0. The precomputed values for G and Q can then be used and the verification performed. The verifier will need 2t/3 point additions, assuming that signed NAF is used to represent a, b and c. The total number of point operations is thus t, which represents a further significant savings compared to 3t/2 with the present invention and without the pre-computed multiple of Q such as described in Given a pre-computed multiple of both Q and G, then uG+vQ can be computed with (t/2)+4(t/4)=3t/2 point operations using conventional representations. When pre-computed multiples of Q are feasible, then the signing equation, in the form above, again provide a significant benefit. The analyses above would be slightly modified when signed binary expansions are used. With yet other known advanced techniques for computing linear combinations of points, some of which are discussed below, the use of the relationship allows signature verification to take up to 40% less time. When implementing scalar multiplication and combinations, it is common to build a table at run-time of certain multiples. These tables allow signed bits in the representation of scalar multiple to be processed in groups, usually called windows. The table costs time and memory to build, but then accelerates the rest of the computation. Normally, the size of the table and associated window are optimized for overall performance, which usually means to minimize the time taken, except on some hardware implementation where memory is more critical. A full description and implementing algorithms for such techniques is to be found in Such run-time tables, or windowing techniques, for scalar multiplication techniques can be combined with the revised verification equation in the embodiments described above. When using such tables, the savings are approximately the same as outlined above. The reason the savings are similar is the following simple fact. Tables reduce the number of adds, by pre-computing certain patterns of additions that are likely occur repeatedly, whereas the use of the revised verification relationship reduces the number of doubles by providing for more simultaneous doubling. In fact, when using tables, the number of adds is reduced, so the further reduction of the doubles provided by using the revised verification relationship has even more impact. By way of an example, a common approach to tables, is to use a signed NAF window of size 5. Building a table for such a NAF requires 11 adds. In the example above where the signer sends a pre-computed multiple uQ of Q, the verifier can build tables for R, Q and uQ, at a cost of 33 adds. It is presumed that verifier already has the necessary tables built for G. Using the pre-computed doubles, the verifier only needs t/6 simultaneous additions for verification. These savings improve as the key size increases. For the largest key size in use today, the savings are in the order of 40%. Such tables do of course require the necessary memory Similarly, computation techniques known as joint sparse forms could be used for computational efficiency. As described above, the integers w, z were found using the extended Euclidean algorithm. Alternative iterative algorithms may be used, including the continued fractions approach. In the continued fractions approach, which is essentially equivalent to the extended Euclidean algorithm, one finds a partial convergent γ/δ to the fraction v/n, such that δ is approximately n As noted above, a conventional ECDSA signature, does not include the point R but instead, it includes an integer x′ obtained from r=x mod n, where R=(x, y). The verifier therefore needs to recover R. The method to recover R discussed above is to supplement the signature (r, s) with additional information i. This information can be embedded in the message, for example. The verifier can use r and i to compute R. When p>n, there is a negligible chance that x′>n and consequently r=x−n. If however such a case does occur, the verification attempt will fail. Such a negligible failure rate for valid signatures can be accepted, or dealt with in the following manner. As shown in Other techniques for determining R can be utilized. In non-cyclic curves, there is a cofactor h, which is usually 2 or 4 in practice. This can lead to multiple possible values of x. The probability that r=x is approximately 1/h. In other situations, we will generally have r=x−n (if h=2), or more generally r=x−mn where (m is between 0 and h−1). Because p is approximately hn, then except in a negligible portion of cases there will be h possible values of x that are associated with r. To make recovery of x, and hence R easier, the signer can compute m and send it to the verifier within the message or as a further signature component. Alternatively, the verifier can make an educated guess for m. This can be illustrated in the case of h=2. Corresponding to r is a correct x and a false value x A similar method may be utilized with a cofactor h=4. In fact, a higher value of h reduces the probability of each of the potential x values from being valid. There are more potential x values, but the analysis shows a similar benefit to the verifier. There are three false values of x, and each has a probability of ⅛ of appearing valid with a fast check. The chance that no false values appear to be a valid x with a fast check is thus (⅞) The inclusion of i (and of m if necessary) is quite similar to replacing r by a compressed version of R consisting of the x coordinate and the first hit of the y coordinate. This alternative, of sending a compressed value of R instead of r, has the advantage of being somewhat simpler and not even a negligible chance of false recovery. Accordingly, as shown in To verify the signature, the recipient In some situations, no channel may be available for the signer to send extra bits. For example, existing standards may strictly limit both the signature format and the message format leaving no room for the sender to provide additional information. Signers and verifiers can nevertheless coordinate their implementations so that R is recoverable from r. This can be arranged by the signer, as shown in As an alternative to modifying R as described above, and to maintain strict conformity to the ECDSA standard, the value of s may be modified after computation rather than R. In this case, the signer notes that the value of R does not conform to the prearranged criteria and proceeds to generate r and s as usual. After s is computed, the value is changed to (−s) to ensure the verification will be attained with the presumption of the prearranged value of y. When a signer chooses a signature (r, s) such that R is implicitly recovered, an ordinary verifier will accept the signature as usual. Such signatures are perfectly valid. In other words, the revised verification is perfectly compatible with existing implementations of ECDSA verification. An accelerated verifier expecting an implicitly efficient signature but receiving a normally generated signature, will need to try two different values of i. If accelerated verification takes 60% of the time of a normal verify, then in a cyclic curve (cofactor h=1), the average time to needed verify a normal signature is 50% (60%)+50% (120%)=90% of a normal verify. This because 50% of the time a normal signature will have i=0, requiring just one implicitly accelerated verify, and the other 50% of the time, two accelerated verifies are needed. Thus an implicitly accelerated verify will still be faster than a normal verifier, even when the signatures are not implicitly accelerated. Conventional signatures may also be modified, either by the signer or by a third party, to permit fast verification. In this case the signature is forwarded by a requestor to a third party who verifies the signature using the verification equality. In so doing the value of R is recovered. The signature is modified to include the indicator I and returned to the requestor. The modified signature may then be used in subsequent exchanges with recipients expecting fast verify signatures. The above techniques recover R to permit a revised verification using the relationship −zR+(zu mod n)G+wQ=O. However, where the ECDSA is being verified, the integers w and z may be used without recovery of R as shown in The above examples have verified a signature between a pair of correspondents Another application is implicit certificate verification. Implicit certificates are pairs (P, I), where P is an elliptic curve point and I is some identity string. An entity Bob obtains an implicit certificate from a CA by sending a request value R which is an elliptic curve point to the CA. The CA returns the implicit certificate (P, I) and in addition a private key reconstruction data value s. Bob can use s to calculate his private key. More generally, any entity can use s to verify that the implicit certificate correctly corresponds to Bob's request value R and the CA public key C. This is done by checking the verification equality H(P, I)R+sG=H(P,I)P+C, where H is a hash function. This equation is equivalent to eQ+sG=C, where e=H(P, I) and Q=R−P. The form of this equation is highly similar to the form of the standard ECDSA verification equation. Consequently, the techniques discussed above may be used to provide a means to accelerate verification of this equation. This is done optimally by determining relatively smaller values w and z such that e=w/z mod n, then multiplying the equation through by z to give: wQ+(sz mod n)G−zC=O. Again, the multiple of G is this equation is full size, but generally multiples of G can be pre-computed, so this does not represent a problem. Another variant that takes advantage of this technique is to shorten all three multiples in the ECDSA signing equation. Theoretically, each multiple can be shortened to a length which is ⅔ the length of n (where n is the order of G). One way to achieve this shortening is by solving the short vector lattice problem in 3 dimensions. Algorithms exist for solving such problems. Shortening all three multiples is most useful when no pre-computed multiples of G can be stored, which makes it more efficient to reduce the length of the multiple of G as much as possible. Such techniques are described more fully in Henri Cohen, “A Course in Computational Algebraic Number Theory”, Springer, ISBN 0-387-55640-0. Sections 2.6 and 2.6 describe the LLL algorithm, its application to finding short vectors in lattices, and also mentions Vallee's special algorithm for 3 dimensional lattices. Another application of this technique is the application to a modified version of the Pintsov-Vanstone Signature scheme (PVS) with partial message recovery. A PVS signature is of a triple (r, s, t). Verification of a signature and message recovery from a signature under public Q, with base generator G, is done as follows. The verifier computes e=H(r∥t), where H is a hash function. The verifier then computes R=sG+eQ. Next, the verifier derives a symmetric encryption key K from R. With this, the verifier decrypts r using K to obtain a recovered message part u. The recovered message is some combination of t and u. The signature is valid only if u contains some redundancy, which is checked by verifying that u conforms to some predetermined format. The PVS scheme is part of draft standards IEEE P1363a and ANSI X9.92. In a modified variant of PVS, verification time can be decreased by utilizing integers w and z. The modified variant of PVS is shown in A method to further accelerate signature verification of digital signature, in elliptic curve groups and similar groups is illustrated as follows. The verification of an ECDSA signature is essentially equivalent to confirmation that a linear combination, such as aR+bQ+cG, of three elliptic curve points, equals the point of infinity. One way to verify this condition is to compute the point aR+bQ+cG and then check if the result is the point O at infinity, which is the identity element of the group as described above. This verification can sometimes be done more quickly than by directly computing the entire sum. For example, if a=b=c, then aR+b Q+cG=O if and only if the points R, Q and G are collinear. Checking if points are collinear is considerably faster than adding to elliptic curve points. Collinearity can be checked with just two field multiplication, by the equation (x Similarly, when b=c=0, so that one wishes to verify that aR=O, in principle one does not need to compute aR in its entirety. Instead one could evaluate the a Recursive formula exist, similar to the recursive formulae for division polynomials, for the denominators of sums like aR+bQ+cG, and these can be compute more quickly than the computing the full value of the point aR+bQ+cG. Knowledge of the group order n can further improve verification time. Yet another application of this technique is to provide efficient recovery of the public key Q from only the ECDSA digital signature as shown in Given an ordinary ECDSA signature (r, s), one can recover several candidate points Q that could potentially be the public key. The first step is recover the point R. Several methods have already been described for finding R in the context of accelerated verification, such as: by inclusion of extra information with the signature; by inclusion of extra information in the message signed; by extra work on the signer's part to ensure one valid R can correspond to r; and by extra work on the verifier's part of trying a multiplicity of different R values corresponding to r. Once R is recovered by one of these methods, then the public key Q can be recovered as follows. The standard ECDSA verification equation is R=(e/s)G+(r/s)Q, where e=H(M) is the hash of the message. Given R and this equation, solving for Q is done by Q=(s/r)R−(e/r) G. However, since with a significant probability a pair (r, s) will yield some valid public key, the correspondent Correspondent H will also be appreciated that each of the values used in the verification equality are public values. Accordingly, where limited computing power is available at the verifier it is possible for the signer to compute the values of w and z and forward them with R as part of the message. The recipient then does not need to recover R or compute w and z but can perform the verification with the information available. The verification is accelerated but the bandwidth increased. Although the descriptions above were for elliptic curve groups, many of the methods described in the present invention applies more generally to any group used in cryptography, and furthermore to any other application that uses exponentiation of group elements. For example, the present invention may be used when the group is a genus 2 hyperelliptic curve, which have recently been proposed as an alternative to elliptic curve groups. The above techniques may also be used to accelerate the verification of the Digital Signature Algorithm (DSA), which is an analogue of the ECDSA. Like ECDSA, a DSA signature consists of a pair of integers (r, s), and r is obtained from an element R of the DSA group. The DSA group is defined to be a subgroup of the multiplicative group of finite field. Unlike ECDSA, however, recovery of R from r is not easy to achieve, even with the help of a few additional bits. Therefore, the present technique applies most easily to DSA if the value is R sent with as part of the signed message, or as additional part of the signature, or as a replacement for the value r. Typically, the integer r is represented with 20 bytes, but the value R is represented with 128 bytes. As a result, the combined signature and message length is about 108 bytes longer. This could be a small price to pay to accelerate verification by 33%, however. In the DSA setup, p is a large prime, and q is smaller prime and q is a divisor of (p−1). An integer g is chosen such that g^q=1 mod p, and 1<g<p. (Note that q and g correspond to n and G, respectively, from ECDSA.) The private key of the signer is some integer x and the public key is Y=g^x mod p. The signer generates a signature in the form (R,s) instead of the usual (r, s). Here, R=g^k mod p, whereas, r=R mod q. In both cases, s=k^(−1)(h(M)+x r) mod q, where x is the private key of the signer, M is the message being signed, and h is the hash function being used to digest the message (as in ECDSA). In normal DSA, the verifier verifies signature (r, s) by computing u=h(M)/s mod q and v=r/s mod q, much like the u and v in ECDSA embodiments, and then checks that r=(g^u Y^v mod p)mod q. In this embodiment, the verifier finds w and z of bit length about half that of q, so that each of w and z is approximately sqrt(q), such that v=w/z mod q. This is done by the same method as in ECDSA embodiment above, with n replaced by q. The verifier then computes:
If this quantity equals 1, then verifier accepts the signature, otherwise the signature is rejected. The verifier computes this quantity using the square-and-multiply algorithm, or some variants thereof, and exploits simultaneous squaring, which is analogous to simultaneous doubling in ECDSA. Many of the methods of ECDSA fast verify may be used for DSA fast verify. A pre-computed multiple of the g, say j, may be used, so that the computation looks like: R^z g^S j^t Y^W mod p where each of z, s, t and w has bit length about half that of q. If pre-computed powers of the public Y are made available, then the lengths of the exponents can be further reduced, thereby further reducing the number of doublings, making the verification yet faster. Patent Citations
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